CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS

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1 Bu Korean Math Soc 50 (2013), No 1, pp CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS Zhongping Li, Chunai Mu, and Wanjuan Du Abstract In this paper, we consider the positive soution to a Cauchy probem in of the fast diffusive euation: x m u t = div( u p 2 u)+ x n u, with nontrivia, nonnegative initia data Here 2 +1 < p < 2, > 1 and 0 < m n < m+n( 1) We prove that c = p 1+ p+n is the critica Fujita exponent That is, if 1 < c, then every positive soution bows up in finite time, but for > c, there exist both goba and non-goba soutions to the probem (11) 1 Introduction In 1966, Fujita [4] considered the foowing initia vaue probem u t = u+u p, x, t > 0, u(x,0) = u 0 (x), x, where N 1,p > 1 and u 0 (x) is a bounded positive continuous function They proved that the probem (11) does not have any nontrivia, nonnegative goba soution if 1 < p < p c = N, whereas if p > p c, there exist both goba and bowing up soutions Such a number p c is then caed to be the critica Fujita exponent In [10, 29], Hayakawa and Weisser have shown that p c = 1+ 2 N beongs to the bow-up case These eegant works reveaed a new phenomenon of noninear PDEs and stimuated the study of simiar features for various noninear evoution euations (see, eg the survey papers [1, 12] and the references therein, and aso the recent papers [4-11, 13-24, 26-29, 32]) The critica Fujita exponent for the foowing Cauchy probem was given as p c = m+ 2 N (12) u t = u m +u p, x, t > 0, u(x,0) = u 0 (x), x, Received May 12, 2011; Revised October 19, Mathematics Subject Cassification 35K50, 35K55, 35B33 Key words and phrases critica Fujita exponent, fast diffusive euation, variabe coefficients 105 c 2013 The Korean Mathematica Society

2 106 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU where p > 1, m > 1 or 1 > m > (N 2)+ N and u 0 (x) is a bounded positive continuous function In [7, 8, 23] they have proved that the soution u(x, t) of (12) bows up if 1 < p < p c ; whie both goba and nongoba positive soutions exist if p > p c When p c = m+ 2 N, in [18, 19] Mochizuki, Mukai and Suzuki have proved that the soutions of (12) bow up in finite time (see aso [6, 9]) Qi [21] repaced the constant coefficient of the noninear source in (12) by the positive function x σ to get the euation (13) u t = u m + x σ u p, x, t > 0, u(x,0) = u 0 (x), x, and estabished the critica Fujita exponent p c = m+ 2+σ (N 2)+ N for m > N In [26], some Fujita type resuts for (11), (12) and (13) were extended to (14) x m u t = u k + x n u, x, t > 0, with > k 1 and 0 < m n < m+ 1 and the critica Fujita exponent of (14) was given as c = k + 2+n The Cauchy probem of another noninear diffusive euation of the form (15) u t = div( u p 2 u)+ x n u, p > N +1 2N, > 1, was aso considered by some authors For the probem (15) with p > 2 and n = 0, Gaaktionov and Qi [5, 6, 20, 22] obtained that c = p 1+ p N is the critica Fujita exponent of (15) and c beongs to the bow-up case If n 0 in (15), QiandWang[24]provedthatthe criticafujita exponent c = p 1+ p+n N for N+1 2N < p < 2 Recenty, Martynenko and Tedeev [16, 17] studied the Cauchy probems of the foowing two euations with variabe coefficients: ρ(x)u t = div(u m 1 u λ 1 u)+u p, x, t > 0 and ρ(x)u t = div(u m 1 u λ 1 u)+ρ(x)u p, x, t > 0, where λ > 0, m+λ 2 > 0,p > m+λ 1, ρ(x) = x n or ρ(x) = (1+ x ) n It was shown that under some restrictions on the parameters, any nontrivia soutiontothecauchyprobembowsupinafinitetime Moreover,theauthors estabished a sharp universa estimate of the soution near the bow-up point In this paper, we consider the positive soution to the Cauchy probem of a fast diffusive euation with variabe coefficients (16) x m u t = div( u p 2 u)+ x n u, x, t > 0, u(x,0) = u 0 (x), x, where 2 +1 < p < 2, > 1,0 < m n < m + N( 1) and u 0(x) is a nontrivia, nonnegative, bounded and appropriatey smooth function The euation in (16) appears in different modes in non-newtonian fuids, here

3 CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION 107 (16) may be used to describe the temperature u of the channe fow of a fuid; the term x m corresponds to the reciproca of the diffusivity; thus due to m > 0, it hastheeffect ofraisingthetemperaturefor x < 1and decreasingthe temperature for x > 1(see[25, 31], where a more detaied physica background can be found) Since m > 0 and 2 +1 < p < 2, the euation in (16) is fast diffusive, which has the infinite speed of propagation property, and hence the soutions of (16) become instantaneousy positive everywhere That is the reason why we are restricting ourseves to positive soutions in this paper In this work, we are interested in the arge time behavior of soutions of the Cauchy probem (16) and estabish the critica Fujita exponent c = p 1+ p+n Namey, we wi prove that the soutions of the probem (16) have the foowing properties: (i) If 1 < < c, then each positive soution bows up in a finite time (ii) If > c, then there exist both nontrivia goba and nongoba soutions with sma and arge u 0, respectivey Unfortunatey, we assume p > 2N 2 N+1 instead of p > +1 when we prove that c = p 1+ p+n beongs to bow-up case In this case we are not abe to cose the gap appearing forn > 1betweenp = 2 2N 2 +1 andp = N+1 since +1 < 2N N+1 whenn > 1; we beieve, however, that this is mainy due to technica difficuties, and that for N > 1 the behavior is actuay the same as for N = 1 Wesaythat u(x,t) isthe soutiontothe probem(16)in Q T = (0,T)if u(x,t) C(Q T ), Du L 1 1oc (0,T;L1 1oc (RN )) and (16) is satisfied in the sense of distribution in Q T, where T > 0 is the maxima existence time The oca existence in time and uniueness of soutions and the comparison principe for the probem (16) can be found in [2, 30] In this context, the soution u(x,t) is caed bow-up in finite time T > 0 if w(t) = Ωu(x,t)dx + as t T for a finite T > 0, where Ω is a bounded domain in It foows easiy that this definition is the same as that of Friedman and McLeod [3] since u(x,t) is a continuous function in Q T When we consider the bow-up case, without oss of generaity, we assume that u 0 (x) is radiay symmetric and nonincreasing, ie, u 0 (x) = u(r) with r = x, andu 0 (r) isnonincreasinginr Therefore, bythecomparisonprincipe, we have that the soution of (16) is aso radiay symmetric and non-increasing in r = x Since the soutions of (16) become instantaneousy positive everywhere, we can assume that initia data u 0 (x) is a positive function If u 0 (x) is not radiay symmetric and non-increasing, we consider the (nonincreasing) soution v to (16) corresponding to the initia vaue v 0 (x) = inf{u 0 (y),0 y x }, which is radiay symmetric and nonincreasing in r If v bows up in finite time, so does u This paper is organized as foows In Section 2 we estabish the critica Fujita exponent c and the critica case = c is investigated subseuenty in Section 3

4 108 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU 2 Critica Fujita exponent In this section we give the critica exponent of Fujita type That is, we sha show when a soutions of (16) bow up in a finite time or both goba and nongoba soutions exist Theorem 21 If 1 < < p 1 + p+n, then every nontrivia nonnegative soution of the probem (16) bows up in finite time Proof Let 1, 0 x 1, φ(x) = cos 1 2 ( x 1)π, 1 < x < 2, 0, x 2 It foows that for > 0, φ (x) = φ( x ) is a smooth, radiay symmetric, and non-increasing function which satisfies φ C, φ C 2, φ φ C 2 for < x < 2, where and in the seue C is used to represent positive constant independent of, and may change from ine to ine Mutipying the euation in (16) by φ and integrating it over, we obtain (21) d dt x m uφ dx = div( u p 2 u)φ dx+ x n u φ dx R N 2 a(n) u p 1 φ rn 1 dr+ x n u φ dx, here a(n) denotes the voume of the unit ba in Using Höder s ineuaity, we have (22) 2 u p 1 φ r N 1 dr ( ) p 1 ( 2 2 u φ r N 1 dr φ r dr) N 1 Note that φ ν = 0 on B and φ ν 0 on B 2, where B is the ba in with radius and center at the origin, and ν is the outward norma vector on the boundary B or B 2 Since u and φ are radiay symmetric, and non-increasing functions, our empoying poar coordinates gives 2 u φ rn 1 dr = 1 u φ dx a(n) B 2 \B 1 u φ a(n) B 2 ν ds u φ dx B 2 \B = 1 u φ a(n) ν ds (23) u φ dx B 2 \B (B 2 \B )

5 CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION 109 = 1 a(n) 1 a(n) u φ dx B 2 \B u φ dx B 2 \B We continue to by using Höder s ineuaity to discover u φ dx B 2 \B (24) ( ) 1 ( ) 1 x n u φ dx ( x n φ φ 1 ) 1 1 dx B 2 \B B 2 \B Next by a direct computation we have (25) ( 2 φ dr) rn 1 = C (N 1)(), ( ) 1 ( x n φ φ 1 ) 1 1 dx B 2 \B = C N( 1) n 2 Combining (22)-(25), we find (26) 2 u p 1 φ rn 1 dr C (N 1)()+[N( 1) n 2](p 1) ( x n u φ dx Set w (t) = x m uφ dx, t > 0 Substituting (26) into (21) yieds (27) dw dt = ( x n u φ dx ) p 1 )p 1 ( C (N 1)()+[N( 1) n 2](p 1) ) +( x n u φ dx) p+1 By the Höder s ineuaity with m n > N( 1), we have ( x m uφ dx x m n 1 φ dx ( C N+n ) 1 ( x n u φ dx x n u φ dx ) 1, ) 1

6 110 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU ie, (28) x n u φ dx Cw N m+n+n According to (27) and (28), we obtain (29) dw dt ( N m+n+n)(p 1) w p 1 ( C 1 (N 1)()+[N( 1) n 2](p 1) +C 2 w p+1 Under the assumption < p 1+ p+n, we see (N 1)()+ [N( 1) n 2](p 1) < ) ( N m+n+n)( p+1) ( N m+n +n)( p+1) Thus using the fact that w is an increasing function of, we find from (29) that, for sufficienty arge, there exists a constant δ > 0 such that dw δw dt, t > 0 Recaing that supp φ = B 2, we then foow w, and conseuenty u, bows up in finite time since > 1 Theorem 22 If > p 1+ p+n, then the soution of the probem (16) with appropriatey arge initia data bows up in finite time Proof We know from the proof of Theorem 21 that u satisfies (29) If u 0 is sufficienty arge such that, for some > 0, C 1 (N 1)()+[N( 1) n 2](p 1) Then, from (29), we see that and dw dt ( N m+n+n)(p 1) w p C 2w p+1 (0) ( N m+n+n)( p+1) w (t) w(0), t > 0 ( 1 2 C 2w p+1 ( N m+n+n)( p+1) ) δw, t > 0, for some δ > 0 This verifies that u bows up in finite time Theorem 23 If > p 1 + p+n, then the probem (16) admits goba soutions with sma initia data Proof We investigate the auxiiary function (210) u(x,t) = (t+1) α f(ξ), ξ = x (t+1) β, where f(ξ) is to be determined ater and α = p+n m( p+1)+n(p 2)+p( 1), β = p+1 p+n α

7 CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION 111 Then we have u t = (t+1) (α+1) ( αf(ξ) βξf (ξ)), u p 2 u = (t+τ) (p 1)(α+β) f p 2 f (ξ) x, div ( u p 2 u ) = (t+1) (p 1)α pβ ( ( f p 2 f ) (ξ)+ N 1 ξ ) f p 2 f (ξ) u(x,t) is a supersoution of the probem (16) with sma initia data u 0 if the function f(ξ) satisfies (211) ( f p 2 f ) (ξ)+ N 1 f p 2 f (ξ)+βξ m+1 f (ξ)+αξ m f(ξ)+ξ n f (ξ) 0, ξ ξ > 0, f(0) > 0, f (0) = 0 Take (212) f(ξ) = (a+aξ p+m p 1, where a, A > 0 are constants to be chosen After a computation, we have f (ξ) = p+m p+m A(a+Aξ 1 m+1 ξ p 1, (f p 2 f (ξ) = ( p+m )p 1 A p 1 (a+aξ p+m p 1 ξ m+1, ( f p 2 f ) (ξ) = ( p+m )p A p (a+aξ p+m 1 p(m+1) ξ p 1 (m+1)( p+m )p 1 A p 1 (a+aξ p+m p 1 ξ m Then inserting the expression of f(ξ) into the first ineuaity in (211), we obtain namey, (213) p+m p+m A(a+Aξ 1 p(m+1) ξ p 1 [( p+m )p 1 A p 1 β] +(a+aξ p+m p 1 ξ m [α (N +m)( p+m )p 1 A p 1 ] +ξ n (a+aξ p+m (p 1) 0, p+m p+m p 2 p+m p+m A(a+Aξ ξ p 1 [( )p 1 A p 1 β] +[α (N +m)( p+m )p 1 A p 1 ] +ξ n m (a+aξ p+m (p 1)( 1) 0

8 112 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU Noting that > p 1+ p+n α impies that β >, we choose A such that α N +m ( p+m )p 1 < A p 1 < β( p+m )p 1 and define By taking a A for every ξ > 0, we have σ = (N +m)( p+m )p 1 A p 1 α (n m)() (p+m)() (p+m)( 1) (n m)() σ [(p+m)( 1) (n m)()](p 1), ξ n m (a+aξ p+m (p 1)( 1) σ Thus, with this choice of a,a, from (213) we check that the first ineuaity of (211) is vaid Thus, for the case > c, we have constructed a cass of goba sef-simiar supersoutions defined by (210) and (212) Owing to the comparison principe, the soution of the probem (16) is goba if the initia data u 0 is sma enough 3 The critica case = c In order to study the critica case = c, we give a arge time behavior of the soution to (16) for x > 1 Lemma 31 The positive soution of the probem (16) has, for each t (τ,t), (31) u(x,t) ǫ(t τ) α (1+δr k ) γ, x > 1, where T is the maxima existence time for the soution, ǫ,δ are positive constants and N +m α = (p 1)(N +m)+p N, β = 1 (p 1)(N +m)+p N, k = p+m p 1, γ = p 1, r = x (t τ) β Proof Our idea is to show that any positive soution of the probem (16) is, for x > 1, bigger than the foowing simiarity soution where U(t,x) = t (p 1)()+p N U λ (t,x) = λ p+m U(t,λx), (1+b x p+m ) p+m p 1 p 1 t [(p 1)()+p N](p 1) with b = p+m β 1 p 1 Let 0 < τ < T < T and S = [τ,t ] (1,+ ) Since the positive soution u(x,t) is continuous in (0,T ] [0,+ ), there exists δ = δ(τ,t ) > 0 such that (32) δ = minu(x,t), τ t T, 0 x 1

9 CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION 113 We now seect λ > 0 such that (33) U λ (t τ,x) δ, τ t T, x 1 2 To this aim, according to the definition of U λ (t,x) we need (34) (t τ) (p 1)()+p N (1+bλ p+m p+m p 1 x p 1 (t τ) λ p+m or δ p 2 p 1 λ p+m p 1 (t τ) ()() p+m [(p 1)()+p N](p 1) +b x for τ t T and x 1 2, which is impied by (35) δ p 2 p 1 λ p+m p 1 (t τ) ()() p+m [(p 1)()+p N](p 1) p 1 (t τ) [(p 1)()+p N](p 1) +b( 1 )p+m p 1 2 (t τ) ) p 1 δ, p+m ()() [(p 1)()+p N](p 1) p+m ()() [(p 1)()+p N](p 1) Sincetheright-handsideof(35)isboundedbeowbycλ p+m ()() p 1, where c is a constant independent of λ, (35) is satisfied if we choose λ such that λ cδ p+m ()() Since x m U λt = div( U λ p 2 U λ )in S andu λ (t τ,x) = 0 for t = τ, x 1, by (32), (33) and the comparison principe we have U λ (t τ,x) u(x,t), τ < t < T, x 1 Hence, the estimate (31) hods by etting T tend to T Theorem 32 If = p 1+ p+n of (16) bows up in finite time 2N and p > N+1, then every positive soution Proof Assume by contradiction that u(x, t) is a non-trivia goba soution of the probem (16) Noticing that = p 1+ p+n impies (N 1)()+ [N( 1) n 2](p 1) From (29), we find that (36) C 2 w p+1 2C 1,,t > 0, = ( N m+n +n)( p+1) where C 1 and C 2 are independent of and t Otherwise, we can prove w, and hence u, bows up in finite time Noting that im w (t) = x m udx, + by (36) we obtain that (37) x m udx C, t > 0

10 114 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU Recaing (21)-(23) and (25), we deduce for > 1, d x m uφ dx dt ( p 1 C (N 1)() u φ dx) + x n u φ dx B 2\B ( ) p 1 = C (N 1)() φ uφ dx + x n u φ dx B 2\B φ (38) ( p 1 C (N 1)() 2(p 1) uφ dx) + x n u φ dx B 2\B ( p 1 C (N 1)() 2(p 1) x m uφ dx) + x n u φ dx B 2\B p 1 C (R (N 1)() 2(p 1) x m uφ dx) + x n u φ dx N Considering the assumptions p > 2N N+1, we know (N 1)() 2(p 1) < 0 Taking + in (38), from (37) we obtain d dt Let w(t) = x m udx, then x m udx 1 2 (39) w(t) w(0) 1 2 t Empoying Lemma 31 we have x n u (x,t)dx ǫ (t τ) 1 0 c(t τ) 1 x n u dx x n u (x,s)dxds Then by (39) we have im w(t) = +, t + ie, im x m udx = + t + This is a contradiction to (37) y (t τ) β y n (1+δ y k ) γ dy Acknowedgements This work was partiay supported by NNSF of China ( ), partiay supported by Projects Supported by Scientific Research Fund of SiChuan Provincia Education Department(09ZA119)

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12 116 ZHONGPING LI, CHUNLAI MU, AND WANJUAN DU [24] Y W Qi and M X Wang, Critica exponents of uasiinear paraboic euations, J Math Ana App 267 (2002), no 1, [25] A E Scheidegger, The Physics of Fow Through Porous Media, third ed, Buffao, Toronto, 1974 [26] C P Wang and S N Zheng, Critica Fujita exponents of degenerate and singuar paraboic euations, Proc Roy Soc Edinburgh Sect A 136 (2006), no 2, [27] C P Wang, S N Zheng, and Z J Wang, Critica Fujita exponents for a cass of uasiinear euations with homogeneous Neumann boundary data, Noninearity 20 (2007), no 6, [28] Z J Wang, J X Yin, C P Wang, and H Gao, Large time behavior of soutions to Newtonian fitration euation with noninear boundary sources, J Evo Eu 7 (2007), no 4, [29] F B Weisser, Existence and nonexistence of goba soutions for a semiinear heat euation, Israe J Math 38 (1981), no 1-2, [30] Z Q Wu, J N Zhao, J X Yin, and H L Li, Noninear Diffusion Euations, Word Scientific, Singapore, 2001 [31] Y B Zedovich and Y P Raizer, Physics and shock waves and high-temperature hydrodynamic phenomena, Moscow, vo 2, Academic Press, New York, 1967 [32] S N Zheng and C P Wang, Large time behaviour of soutions to a cass of uasiinear paraboic euations with convection terms, Noninearity 21 (2008), no 9, Zhongping Li Coege of Mathematic and Information China West Norma University Nanchong , P R China E-mai address: zhongping-i@hotmaicom Chunai Mu Coege of mathematics and Statistics Chonging University Chonging, , P R China E-mai address: chunaimu@yahoocomcn Wanjuan Du Coege of Mathematic and Information China West Norma University Nanchong , P R China E-mai address: duwanjuan28@163com